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袁萌  陈启清  12月3日

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Tarski’s Truth Definitions

First published Sat Nov 10, 2001; substantive revision Mon Aug 20, 2018

In 1933 the Polish logician Alfred Tarski published a paper in which he discussed the criteria that a definition of ‘true sentence’ should meet, and gave examples of several such definitions for particular formal languages. In 1956 he and his colleague Robert Vaught published a revision of one of the 1933 truth definitions, to serve as a truth definition for model-theoretic languages. This entry will simply review the definitions and make no attempt to explore the implications of Tarski’s work for semantics (natural language or programming languages) or for the philosophical study of truth. (For those implications, see the entries on truth and Alfred Tarski.)

1. The 1933 programme and the semantic conception

1.1 Object language and metalanguage

1.2 Formal correctness

1.3 Material adequacy

2. Some kinds of truth definition on the 1933 pattern

2.1 The standard truth definitions

2.2 The truth definition by quantifier elimination

3. The 1956 definition and its offspring

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1. The 1933 programme and the semantic conception

In the late 1920s Alfred Tarski embarked on a project to give rigorous definitions for notions useful in scientific methodology. In 1933 he published (in Polish) his analysis of the notion of a true sentence. This long paper undertook two tasks: first to say what should count as a satisfactory definition of ‘true sentence’ for a given formal language, and second to show that there do exist satisfactory definitions of ‘true sentence’ for a range of formal languages. We begin with the first task; Section 2 will consider the second.

We say that a language is fully interpreted if all its sentences have meanings that make them either true or false. All the languages that Tarski considered in the 1933 paper were fully interpreted, with one exception described in Section 2.2 below. This was the main difference between the 1933 definition and the later model-theoretic definition of 1956, which we shall examine in Section 3.

Tarski described several conditions that a satisfactory definition of truth should meet.

1.1 Object language and metalanguage

If the language under discussion (the object language) is

L

L

, then the definition should be given in another language known as the metalanguage, call it

M

M

. The metalanguage should contain a copy of the object language (so that anything one can say in

L

L

can be said in

M

M

too), and

M

M

should also be able to talk about the sentences of

L

L

and their syntax. Finally Tarski allowed

M

M

to contain notions from set theory, and a 1-ary predicate symbol True with the intended reading ‘is a true sentence of